Introduction to Applied Algebra: Book Review of Chapter 8-Linear Equations (System of Linear Equations) (original) (raw)
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Solving a system of linear equations (or linear systems or, also simultaneous equations) is a common situation in many scientific and technological problems. Many methods either analytical or numerical, have been developed to solve them so, in this paper, I will explain how to solve any arbitrary field using the different – different methods of the system of linear equation for this we need to define some concepts. Like a general method most used in linear algebra is the Gauss Elimination or variation of this sometimes they are referred as “direct methods “Basically it is an algorithm that transforms the system into an equivalent one but with a triangular matrix, thus allowing a simpler resolution, Other methods can be more effective in solving system of the linear equation like Gauss Elimination or Row Reduction, Gauss Jordan and Crammer’s rule, etc. So, in this paper I will explain this method by taking an example also, in this paper I will explain the Researcher’ works that how t...
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The publication is intended for the Bachelor of Technical and Natural Sciences students. It aims to provide the necessary theoretical knowledge and the different methods on how to solve the systems of linear equations. Examples of solutions for practicing theoretical knowledge are included in this version. Additionally, the publication contains a chapter dealing with the resolution of the examples using the MATLAB and SageMath Systems. The database with the main terms of the subject is provided in the Register of this publication. * Keywords: the system of linear equations, determinant, regular matrix, inverse matrix, Gauss-Jordan elimination, the rank of a matrix, the linear combination of vectors, the linear dependence of vectors, infinitely many solutions, no solution, linear transformations, Computer Algebra, Matlab, SageMath, Bottleneck Algebra. ** Publication type: Textbook, University publication, e-book (online) version. *** Subject area: Mathematics - Linear Algebra - elementary level. Solved tasks of the publication: https://www.mathworks.com/matlabcentral/fileexchange/74889-linear-algebra-a-collection-of-tasks-in-matlab
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Student/Class Goal Students thinking about continuing their academic studies in a post-secondary institution will need to know and be able to do problems on solving systems of equations. Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method.
In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by since it makes all three equations valid. [1] In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system. Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed. See, for example, integer linear programming for integer solutions, Gröbner basis for polynomial coefficients and unknowns, or also tropical geometry for linear algebra in a more exotic structure.
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The Significance of linear system of equations has many problems in engineering and different branches of sciences. To solve the system of linear equations, there have many direct or indirect methods. Gaussian elimination with backward substitution is one of the direct methods. This method is still the best to solve the linear system of equations. Gauss-Jordan method is a simple modification of Gaussian elimination method. Iterative improvement method is used to improve the solution. Our aim is to solve a large system developing several types of numerical codes of different methods and comparing the result for better accuracy. We also represent some practical field where the system of linear equations is applicable.
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World Academy of Science, Engineering and Technology, International Journal of Mathematical and Computational Sciences, 2015
In this paper linear equations are discussed in detail along with elimination method. Guassian elimination and Guass Jordan schemes are carried out to solve the linear system of equation. This paper comprises of matrix introduction, and the direct methods for linear equations. The goal of this research was to analyze different elimination techniques of linear equations and measure the performance of Guassian elimination and Guass Jordan method, in order to find their relative importance and advantage in the field of symbolic and numeric computation. The purpose of this research is to revise an introductory concept of linear equations, matrix theorey and forms of Guassian elimination through which the performance of Guass Jordan and Guassian elimination can be measured.